Exact solution of the boundary-dissipated transverse field Ising model: Structure of the Liouvillian spectrum and dynamical duality

We study the boundary-dissipated transverse field Ising model described by a Lindblad master equation and exactly solve its Liouvillian spectrum in whole parameter space. By mapping into Su-Schrieffer-Heeger with imaginary boundary potentials under parity constraint, we rapidity analytically thus construct strictly constraint condition. Our results demonstrate that displays four different structures, which are characterized numbers of segments. analyzing properties spectrum, can determine phase boundaries between structures prove gap fulfilling duality relation weak strong dissipation region. Furthermore, unveil existence dynamical duality, i.e., long-time relaxation dynamics exhibits almost same behavior region as long holds true.